The category \(\)  is complete and cocomplete. Moreover, the forgetful
functor \(\rightarrow \)  admits both a left and a right adjoint and thus
preserves limits and colimits.


Since the category \(\)  is complete and cocomplete, with products (coproducts) being given by supremum (infimum), by general results about
enriched categories [1]}, the category
\(=\)  has small limits and colimits. The left (right)
adjoint to the forgetful functor \(\rightarrow \)  is the functor which to a
set \(X\)  associates the generalized metric space \((X,d_0)\)  (\((X,d_\infty )\) ). The underlying set of a limit of metric spaces is thus the
limit of the underlying sets and similarly for colimits.
